Infrared singularities, vacuum structure and pure phases in local quantum field theory

A NNALES DE L ’I. H. P., SECTION A

G. M ORCHIO

F. S TROCCHI

Infrared singularities, vacuum structure and pure phases in local quantum field theory

Annales de l’I. H. P., section A, tome 33, n

^o

3 (1980), p. 251-282

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251

Infrared singularities,

vacuum

structure and _pure phases

in local quantum ^field theory

G. MORCHIO

F. STROCCHI

Istituto di Fisica dell’Università, Pisa, Italy

Scuola Normale Superiore ^and INFN, Pisa, Italy

Vol. XXXIII, ~3.1980. Physique ^’ théorique. ^’

ABSTRACT. The occurrence of infrared

singularities

^{of the}

confining

type

imply

^{that the} associated quantum ^field

theory

^cannot

satisfy

^the

positivity

condition and therefore one has a strong

departure

from standard

(positive metric) QFT’s.

^The

general

^structure

properties

of indefinite metric local

QFT’s (which

^include gauge

QFT’s)

^are

investigated.

^In

parti-

cular we discuss the

problem

^of

associating

^{a Hilbert} space structure to

a

given

^{set of}

Wightman functions,

^the

properties

of maximal Hilbert space structures, ^the connection between the occurrence of infrared

singu-

larities and the existence of ^more than one translation invariant state

(e-vacua),

^the definition and

uniqueness

^{of the vacuum state} and its relation with the

irreducibility

of the local field

algebra (pure phases).

RESUME. ^- L’existence de

singularites infrarouges

^du type ^confinant

implique

que la theorie des

champs

^{associee ne} peut pas satisfaire la condi- tion de

positivité

^{et par}

consequence

^{on a un fort}

depart

par rapport ^aux

TCQ’s

^standards

(metrique positive).

^Les

proprietes generates

^{de structure}

des theories des

champs quantiques

ayant ^une

metrique

^indefinie

(qui

comprennent les theories des

champs de jauge)

^{sont étudiées. En}

particulier

nous discutons Ie

probleme

d’associer une structure Hilbertienne a ^un

l’Institut Henri Poincaré-Section A-Vol. XXXIII, 0020-2339/1980/251 $5,00/

~ Gauthier-Villars

252 G. MORCHIO AND F. STROCCHI

ensemble donne de fonctions de

Wightman,

^les

proprietes

^{des structures}

Hilbertiennes

maximales,

la connection entre 1’existence de

singularites infrarouges

^et 1’existence de

plus qu’un

^{etat invariant par} translations

(0-vacua),

la definition et l’unicité de l’état du vide et sa relation avec l’irrédu- cibilite de

l’algèbre

locale des

champs (phases pures).

1. INTRODUCTION

For a

long

^{time the main}

problem

^{of local} quantum ^field

theory (QFT)

has been the control and the elimination of ultraviolet

divergences,

^first

as a crucial

practical problem

^for

computing

^finite

higher

order contri- butions in

perturbation theory [1] ] and

^{then as a}

question

^of whether the

theory

^was

internally

consistent

[2] ]

^and

mathematically acceptable [3 ].

With the advent of _gauge

QFT’s [4] it

^{has been}

suggested

^{that it}

might

be better to have a

good

ultraviolet behaviour at the

price

^{of a} bad infrared structure, ^{with the}

hope

^{that the infrared}

problem

would be solved

by

^a

correct identification of the

physical

^states

(confinement mechanism).

^The

properties

of such theories has been the

subject

^of many

investigations during

the last years both at the level of the

perturbation theory [4 ],

^with

improvements

which take the non linear effects of the classical solutions into account

[5 ],

^{and at the} level of constructive

QFT [6 ]

^{in the lattice}

field

theory approach.

^{It seems however that the main infrared}

problems,

connected with the construction of the ^«

charged »

^{states, are still} open, apart ^{from the abelian case}

[7].

^The

deep physical

^{reason is that in such}

theories there exist «

phases »

^{or « sectors » which cannot be} characterized in terms of

expectation

values of local observables or

by

local order _para- meters, but one must use observables of the type ^of

charges

^which

obey

^a

Gauss’ law

[8 ],

^and characterize such sectors

by

^the

expectation

^{value of a}

loop

^{or a flux at}

infinity [5 ].

A direct consequence of such

phenomena

is that not all the

physical

^{states, in}

particular

^the

charged

^{states and in}

general

^{states with non trivial «}

topological »

numbers like the 0-vacua

[5 ],

cannot be described in terms of local, excitations of the vacuum

[9 ], [8] ]

nor

by

^local

morphisms

^{of the}

algebra

of observables.

Therefore,

^a

by

^far

non trivial step ^{is involved in}

going

from the Green’s ^or

Wightman’s

functions of the local fields to the construction of all the

physical

states, in

particular

^the

charged

^states.

From a more technical

point

^of

view,

the above difficulties are

strictly

related to the fact that such theories cannot be described in terms of an

irreducible set of local fields without

giving

up the

positivity

^condition

and one is rather

naturally

^{lead to} indefinite metric

QFT’s.

^The

necessity

^of

Annales de l’Institut henri Poincaré-Section A

indefinite metric can be

proved quite generally

when there are

charges

which

obey

^{a Gauss’law}

[9 ] [8] ] (gauge QFT’s)

and it is also unavoidable whenever the Green’s or

Wightman’s

functions exhibit infrared

singula-

rities associated with a «

confining » potential,

^{since such}

singularities

^are

incompatible

^with

positivity,

^{in a local}

QFT [10 ].

^{In all such cases we have}

a strong

departure

from standard

(positive metric) QFT [77] ]

^{where the} quantum mechanical

interpretation

^{of the}

theory, equivalently

^{the identi-}

fication of the

physical

^{states is}

uniquely

^fixed

by

the local states, whereas in indefinite metric

QFT apparently

^a

large

arbitrariness is involved.

In Sect.

2,

^{3 we}

identify

^{the condition}

(Hilbert

space ^structure

condition)

which

replaces

^{the axiom of}

positivity

and allows the construction of a

Hilbert _space associated to the

given

^set

of Wightman

^functions

(reconstruc-

tion theorem for indefinite metric

QFT’s).

^One of the main

points

^{of our}

analysis

^{is to}

emphasize

that when the

Wightman

functions do not

satisfy

the

positivity condition,

^one may associate with them different Hilbert _space structures,

leading

ⁱⁿ

general

^to

completely

^different spaces of states.

In

particular,

very

important

^structure

properties

like the existence of

more than one translation invariant state

(mixed phase),

spontaneous symmetry

breaking,

existence of

03B8-vacua, reducibility

^{of the field}

algebra, crucially depend

^{on the Hilbert} space ^{structure one} associates to the

given Wightman

^functions.

Among

^{all the}

possible

^Hilbert space structures, it is shown that maximal Hilbert _space structures

(Krein spaces) identify

^{maximal sets of states}

to the

given Wightman

^functions and exhibit _very

important properties.

In

particular,

ⁱⁿ maximal Hilbert _space structures one may establish a

connection between the occurrence of infrared

singularities

^{and the existence}

of more than one translation invariant state

(e-vacua) (Sect. 4),

^a

pheno-

menon which is not

governed

ⁱⁿ

general by

^{the cluster} property ^{as in the} standard

(positive metric)

^case

[12 ].

As illustrative

examples

^{the massless}

scalar field in two dimensions and the

dipole

field in four dimensions are

discussed.

The

question

^{of the}

irreducibility

^{of the field}

algebra (pure phases)

ⁱⁿ

connection with the

cyclicity

^and

uniqueness

^{of the vacuum} is solved in Sect. 5. We also

analyze

^the definition and

uniqueness

^{of the vacuum,}

a concept ^which

requires

^{much more care} when the space time translations

are not described

by unitary

operators ^{and one cannot associate with them}

spectral projectors,

ⁱⁿ

general (Sect. 5). Many

^{of the} basic results of standard

(positive metric) QFT [77] are generalized

^{to the case} of indefinite metric.

In our

opinion,

^the

emerging picture

is that the

general

^{structure of indefi-}

nite metric

QFT’s

is much richer than standard

QFT’s

^{since it can}

naturally

account for

important phenomenon

^like

confining

^infrared

singularities,

©-vacua,

^{absence of local} order paremeters,

Higgs’ phenomenon,

^{etc. In}

particular

^{the use of an} irreducible set of local fields

(in particular

^the

charged fields)

may allow the construction of the

charged

^states

and/or

^{of other}

Vol. XXXIII, ^n° 3-1980.

254 G. MORCHIO AND F. STROCCHI

topological

^{sectors which are not}

directly

available in terms of the

algebra

of observables and whose construction _appears to be far from trivial

starting

^{from the vacuum sector.}

2.

COVARIANCE,

^LOCALITY

AND SPECTRAL CONDITION

As clarified in the

mid-fifties,

^{a field}

theory

is defined

by

^{a set of Green}

or

Wightman

^functions

[3 ]. Actually,

^{this is} the way ^a field

theory

^{is obtained}

either

by perturbation theory

^or

by

constructive field

theory

^methods.

Therefore an

analysis

^{of the structure}

properties

^{of a}

(positive

^or

indefinite)

quantum ^field

theory (QFT) always

^{reduces to a}

study

^{of its}

Wightman

functions.

In this _paper we will focus our attention on local and covariant

QFT, namely

^to theories whose

Wightman

distributions

[11] ] satisfy

^the

following properties (which

^for

simplicity

^we

specify only

^{in the case of an hermitian}

scalar

field).

I. COVARIANCE

For any Poincare

transformation {a, A}

^the

n-point

^{functions are}

invariant

II. LOCALITY

If xi - xi+1 is spacelike

III. SPECTRAL PROPERTIES

The Fourier transforms ...

qn _ 1)

^{of the} distributions

have support ^{contained in the}

cones ~ ~

^0.

The

hermiticity

conditions reades

For the motivations of I-III we refer to

[11].

Even in the

positive

^metric

case, I-III involve in

general

^an

extrapolation

^with respect ^{to the}

physical requirements,

^{as it must be since a}

Wightman

^field

theory

^{has a richer}

structure than the

algebric

formulation of Araki

Haag

^and

Kastler,

^based

exclusively

^{on the}

algebra

of observables

[13 ].

^{The richer}

Wightman

structure, which involves

physical

^{as well as non}

physical

^fields

(e.

g. the

Annales de Poincaré-Section A

fermion

fields)

has indeed

proved

^{to be an}

advantage

^{in the actual cons-}

truction of field theories. The usefulness of

introducing unphysical

^fields

to allow a

simple

^and

practical

definition of

interactions,

^{with the automatic}

validity

^{of basic}

physical properties

^like

microscopic causality

^and

positivity

of the energy, has been

repeatedly recognized

^{in the} past. ^{For these reasons}

we maintain I-III also in the indefinite metric case. This choice also leads to very

important

^technical

advantages.

^We recall the basic role

played by locality

and covariance in the

development

^{and the} very formulation of renormalization

theory [14 ].

^{Even the recent}

developments

^{built on}

the functional

integral

formulation of

QFT

^are

crucially

^{based on the}

analytic

continuation of the

Wightman

functions from Minkowski to Euclidean _space, a property which relies on I-III.

In the

positive

^{metric case two further}

properties

^{are added to}

I-III, namely positivity

^{and the cluster} property, and these allow to recover a

quantum

mechanical

interpretation

^{of the}

theory

^{in the terms} of states, transition

probabilities

and irreducible field operators. ^{For indefinite} metric

QFT

^this

strategy requires

modifications and the main _purpose of this _paper is to discuss them in detail.

With

only

^{I-III at our}

disposal

we can recover

only

^{a set of vector states}

which have a linear structure

[7~].

THEOREM 1. Given a set of

Wightman

distributions ~

satisfying I-III,

one can construct a linear _{space W} a

sesquilinear form (’,-)

^{on W and}

operator valued distributions

f

^E

!/(~4), acting

^{on W such that}

a)

^{there is a vector}

cyclic

^with respect ^{to the}

polynomial alge-

bra ~

generated by

^{the field} operators

with ( B}Io &#x3E;

&#x3E; 0

b)

^{the field} operators

satisfy locality, namely

if _supp

f

^is

space like

^with respect ^to supp g

c)

^{there is a linear}

representation U(a, A)

of the Poincare _group on

W defined

by (P

^{denotes a}

polynomial)

with ⁼

f (l~-

^{~x -}

a~,

^{so that is a} Poincare invariant vector

d)

^the

sesquilinear

^{form is}

hermitean,

^non

degenerate,

Poincare invariant and

Proof.

^{- We start from} the Borchers

[16] algebra

whose elements

are finite sequences

with

fn

^E

~(L~4n).

^The

Wightman

functional ~ defines a linear functional

on g through

Vol. XXXIII. n° 3-1980.

256 G. MORCHIO AND F. STROCCHI

Denoting by

^{x the tensor}

production ~

We get ^the

following sesquilinear

^{form on ~}

which is hermitean as a consequence of condition

(2.3).

^{To obtain a non}

degenerate

^{form we consider the set L of}

elements g

^{E ~ such that}

L is

clearly

^{a linear} space and it is invariant

by

^left

multiplication

We then define as linear _{space W} the set of

equivalence

^classes

[_f

On W the field operators ^{are defined}

by

-

with

1

⁼

(0, f, 0, ... ). Equation (2.13)

is well defined

since, by

eq.

(2.12),

the

right

hand side does not

depend

^{on the choice}

of g

^{within its}

^equiva-

lence class.

Clearly,

^{the vector}

B}I 0

⁼

[~o ]. Yo

⁼

^{0, 0, ... )}

^{is non}

zero and it is

cyclic

^with respect ^{to ~ .} Furthermore

i. e. eq.

(2. 6)

holds. The

completion

^{of the}

proof

^{then follows}

easily.

3. HILBERT SPACE STRUCTURE CONDITIONS

As we have seen in the

previous

^{section the} definition of a

QFT

^{in terms}

of its

Wightman

^{functions leads to a}

representation

^{in terms of field opera-}

tors and local states in a vector space. In

general,

^{it is not}

guaranteed

^that

one may introduce ^a

(pre)

^Hilbert space ^structure in such a vector space W and

provide

^{W with a} quantum mechanical

interpretation

^{and one}

might envisage

^the situation in which such a quantum mechanical structure will _emerge

only

^{at the level of}

asymptotic

^states.

The need for a convenient Hilbert

topology

^{in order to} define the asympto- tic limit and

strongly

^motivated

physical

considerations suggest ^{to consider} the case in which a quantum mechanical structure and a quantum ^mecha- nical

interpretation

emerges also for field

configurations

^at finite times.

Annales de Henri Poincaré-Section A

As we will see in more detail

later,

^the

physical interpretation

^{of the}

theory (or equivalently

the identification of the

physical

^states associated to the

given Wightman functions) crucially

^{relies on the}

specification

^{of a Hilbert}

space structure. In the standard case

[77] ]

^the

QM

^{structure is}

required

to emerge

already

^{at the} level of local states ; ^more

precisely

^{one assumes}

that local states have a

physical interpretation

^as quantum ^mechanical

states and therefore the

Wightman

^{functions are}

required

^to

satisfy

^the

positivity

condition. This amounts to consider

QFT’s

^whose

physical (phase)

^{structure can} be read off at the local

level,

^{i. e. there} exist local observables which take different values in different

phases ~local

^order

parameters) .

This does not cover the case in which different _pure

phases

^{cannot be}

completely

characterized in terms of

expectation

^{values of local}

observables,

but one must use observables of the type ^of

charges

^which

obey

Gauss’law

[17 ].

^{This is the case of}

phases

^{which are} characterized

by

^the

expectation

^{value of a}

loop

^{or a flux at}

infinity.

^{In all such} cases, ^not all the

physical

^{states can} be described in terms of local excitations of the vacuum, ^nor

by

^local

morphisms [18 ].

^The discussion of such states

requires

the introduction of a space of states the structure of which is related to

the local structure of the

theory (i.

^{e. to the local} space

2:)

^{in a more}

compli-

cated way than in the standard case. One has therefore to solve this

problem

first in order to get ^a

physical interpretation

^{of the}

theory

^{and to exhibit}

its

QM

^structure.

To this _purpose we suggest

adopting

^the

approach

in which such a

Hilbert _space structure is related to the

Wightman

^functions

(in

^{a local}

theory)

and therefore it

provides

^{for a concrete and} natural method for

constructing representations

^{of the}

algebra

of observables which cannot be obtained in terms of local

morphisms.

As a first step ^{we have to}

specify

necessary and sufficient conditions in order that a

given

^{set of}

Wightman

^functions may be

given

^{a Hilbert}

space structure. More

precisely

^{we have to}

specify

^the conditions under which there exists a Hilbert space .~f such that the local vectors are dense in Jf

(f/ = Jf)

^{and the}

Wightman

^{functions can} be written in terms of

an indefinite scalar

product,

^{defined for all vectors of Jf. This means}

that we look for

(Hilbert

space

structure,~

i )

^{a Hilbert} space

Jf,

with scalar

product (-, ’),

such that ~ is dense in Jf

_-

ii)

^{and with the} property that there exists a bounded self

adjoint

operator 11 such that

Vol. XXXIII, n° 3-1980. 10

258 G. MORCHIO AND F. STROCCHI

or,

equivalently,

We have then the

following

^results

[19 ].

THEOREM 2. ^- Given a set of

Wightman

^{functions a} necessary and sufficient condition for the

validity ii)

^is that there exists a

set {pn}

of Hilbert seminorms ?~ defined on

~(I~4n)

^{such that}

Proof.

^{- The}

necessity

of condition

(3.3)

follows from Schwartz

inequa- lity

^with

p,(D

⁼

lIt.

On the other hand if _eq.

(3.3) holds,

^{then one} may introduce a scalar

product

^{in ~ .. _..} 2

where

( , )n

is the Hilbert scalar

product

^defined

by

pn.

Clearly

^,

so that

by

^a redefinition of the scalar

product

^{we can}

satisfy

eq.

(3.2).

Remark. ^- Since

~n

^{is a}

locally

^{convex nuclear}

topological

space, it is

always possible

^to describe its

topology by

Hilbert seminorms. The

non trivial

point

^{is the existence} of Hilbert seminorms which

satisfy

eq.

(3 . 3) ;

this is in

general

^not

guaranteed

unless the

Wightman

functionals

~( f *

^x

g) are jointly

continuous in

[ and g [20].

- - -

THEOREM 3. A sufficient condition for the

validity

^{of condition}

(3.3)

is that the

Wightman

^functions

satisfy

^the

following regularity

condition : when smeared in the variables ... xn the

distributions wn(x1,

^...

have ^an order which is bounded

by

^{a number}

Nj independent

^{of n and}

of the test function _g.

Proof 2014 By

^standard arguments, ^the

validity

^{of the condition of the}

theorem

implies

^{that one can construct Sobolev} type seminorms p~ such that

Annales de ^’ Henri Poincaré-Section A

By using

^the

Hermiticity

conditions one can also find a seminorm such that _{.,~.,,... ~,..~ .~ , _ ,}

Thus, by introducing

^{the new seminorms}

we get

The condition of theorem 3 is also _necessary and sufficient for the field operator ^{to be}

strongly

continuous in

f

^{in the f/}

topology; actually

it can be shown that in this case the weak

continuity implies

^the strong

continuity

^{so that the} condition of theorem 3 must be satisfied if one wants the

continuity

^{of the} matrix elements

(~

^{V’P E}

^~,

^{VO E Jf.}

(Clearly

^the

continuity

of the matrix

elements ( C,

^is

already guaranteed by

^theorem

1).

^The

continuity

^{of the} matrix elements

( C,

^{for E ff}

might

^{be a} redundant and not necessary

requirement.

In

general,

^{a Hilbert} space ^structure

satisfying i), n)

^{defines a self}

adjoint

operator ~ ^which may be

degenerate,

^{i. e.} there may be non zero vectors T E such that , ,~~ ~ , ~ , . ~ ...

However one may

always

^{reduce to the case in}

which ’1

^is

non-degenerate by

^the

following

argument.

Let Then

and if

Po

^{denote the}

projector

^on

.fo,

^{the new scalar}

product

g

E ~,

^still

provides

^{a Hilbert}

majorant of (-,’)

^{and the Hilbert}

space JT obtained

by completion

^of

~

^with respect ^to

( , )’,

still satisfies

i), ii)

and therefore it

provides

^{a Hilbert} space with a non

degenerate

^metric

operator.

In

particular,

^{such removal of the}

degeneracy of ~ automatically

^takes

care of non trivial ideals of the Borchers

algebra ~° arising

^from

specific properties

^{of the}

Wightman functions,

^like

locality

^and

spectral

conditions.

We can now state the axiom which

replaces

^the

positivity

^condition

of the standard case.

I V . HILBERT SPACE STRUCTURE CONDITION

There exists a set of Hilbert

seminorm

^{defined on}

~~(f~~n)

such that

Vol. 3-1980.

260 G. MORCHIO AND F. STROCCHI

(We

^will say that the Hilbert _space structure condition is satisfied in a

strong from ^if pn ^are continuous seminorms on

J(R4n)).

The results discussed in Sect.

2,

^{3 then lead to the}

following

reconstruc- tion theorem.

THEOREM 4

(Reconstruction

^theorem

for indefinite

^metric

QFT). 2014

^Given

a set of

Wightman

^functions

satisfying

^{I-IV one can construct}

a)

^a

separable

^Hilbert space

Jf,

with scalar

product (’, ’)

^{and a metric} operator 1] ^{which is}

bounded,

^self

adjoint

^non

degenerate

b)

^a

representation U(a, A)

^of

P~

ⁱⁿ

Jf,

^{where the} _operators

U(a, A)

have a common dense domain D ~

Do

^{= W and are}

~-unitary [22] i.

^e.

c)

^a translation invariant vector

’Po with (

&#x3E; 0

d)

^{a local}

(hermitian)

^field operator

~’

^E

9( fR4)

^{with a dense}

domain

Do,

^{such that}

Furthermore if the seminorms are invariant under

U(a) (and/or U(A))

then

U(a) (and/or U(A)

^can be extended from W to all ofjf and the extended operators ^are

unitary

operators ^{on Jf.}

Finally,

^{if the seminorms}

Pn

^are continuous seminorms on

~((~4n)~

not

only

^{the matrix}

elements ( C, &#x3E;, 1&#x3E;, ’P E Do

^are

tempered

^distri-

butions,

^since

they

^{are finite sums of}

Wightman functions,

but also the matrix elements

(C, ~(/)~P),

^{’P E}

Do, C

^{E ~’’ are}

tempered

distributions.

As discussed

before,

^the

physical interpretation

^{of a}

theory

^defined

by

a set of

Wightman

^functions

crucially

^{relies on the} introduction of a Hilbert space ^structure and it is natural to ask how much arbitrariness is involved in this

procedure.

In the standard _case, the Hilbert _space structure has an

intrinsic

meaning

^{since it is}

directly given by

^{the set of}

Wightman functions,

via the

positivity

condition. In the indefinite metric _case, such a connection is much less

tight and,

ⁱⁿ

fact,

^{to a}

given

^{set of}

Wightman functions,

^one

may associate

completely

^{different Hilbert} space structures

[2~], leading

to

completely

different space of states

[25 ].

Therefore,

whereas in the standard

(positive metric)

^{case the set of local}

states

uniquely

^{fix their}

closure, "

^the indefinite metric case different closures are

available, corresponding

^to different Hilbert _space

topologies.

A more

systematic investigation

^{of this}

problem

is deferred to a

subsequent

paper. For the present paper it is

enough

^{to introduce the}

following

^notion

of

maximality.

DEFINITION. A Hilbert _space structure

Jf)

associated to a

given

set of

Wightman functions,

^{with non}

degenerate

^metric operator 1], is

Annales de l’Institut //(w/ Poincaré-Section A

maximal if there is ^no other Hilbert _space structure

(11,

^associated

to the

given

^{set of}

Wightman functions,

^{with a non}

degenerate ^metric, operator ~,

^such that Jf is

properly

contained in Jf.

Clearly,

^{since we are} interested in

obtaining

^as much information as

possible

^{from the set of}

Wightman functions {W },

it is natural to look for Hilbert _space structures which are

maximal,

^{i. e. such that}

they

^asso-

ciate

to {W }

^{a maximal set of states. We have then}

THEOREM 5. A Hilbert _space structure

(11, X)

associated to a set of

Wightman functions {W}

is maximal

iff ~-1

is bounded.

LEMMA. Given a Hilbert _space structure

(11, X) with 11-

¹

^bounded,

one may redefine the

metric,

^without

changing Jf,

^{in such a} way that the new

metric ~ satisfies ~2

^{= 1.}

~’roof.

^{- Let}

(’, ’)1

^{1 be the scalar}

product

^defined

by

^Then

we define

(-,’)=(- 1111 !’ ’)1

^{1 so that}

Proof of

^{theorem 5. - Given a Hilbert} space

(1],

with scalar

product ( ~ , ~ ) 1,

^{one can}

always

^{introduce a new Hilbert} space ^structure

(1}, Jf)

in such a way

that ~-1

is bounded. We put

The ^new scalar

product

^{defines a Hilbert}

topology!

^on

W,

which is weaker than the

topology r

^defined

by ( ~ , ~ ) 1

^since

The Hilbert _{space X} is

complete

^with respect ^{to T} is bounded.

Therefore is not bounded the above construction shows that

(1], x)

is not maximal. On the other hand if

1]-1

^is bounded and

(1]b

^{is a}

Hilbert _space structure such that

Jf,

^then

(-, ’ ’)1 ~ C( , ),

^which

implies ^(x, y)

1 ⁼

(x, Ay)

^{with A =}

A*,

^{A ~ 00. It then follows}

that _{- ~} and since

1] - 1

is bounded so is A -1 and the two scalar

products

define the same Hilbert

topology.

2014 By

^the above result

given

^{a Hilbert} space ^structure

(1], Jf)

there is

always

^{a new Hilbert} space ^structure

(~, f)

which is maximal and such that

~ ~ ~ ;

^therefore

given

^{a Hilbert} space ^structure

(1], Jf)

it is

always possible

^{to construct a new}

metric ~

^such

that ~2

^{== 1. Inner}

product

space with the property ^{that the metric}

operator ~

^satisfies

~2

^{== 1}

are also called Krein spaces and

they

^{have been}

extensively

^{studied in the}

literature

[2~] ] [26 ].

The above

maximality

^{condition has a direct} consequence in ^terms of

physical

^{states. In fact}

given

^{a Hilbert} space ^structure

(1],

^{for the}

Vol. XXXIII. n° 3-1980.

262 G. MORCHIO AND F. STROCCHI

physical interpretation

^{of the}

theory

^{one has to extract a}

physical

^vector

space f’ ^c X on which the indefinite inner

product

^is

non-negative.

The

physical

^{states are then} identified with the _rays of the Hilbert _space

Kphys ~ ",

with scalar

product

^induced

by .,.&#x3E;,

^where

~" _ ~ x,

^{x E}

Jf’, ~, ~ ~

⁼

0 }.

^In

general

^a

completion

^is necessary since

might only

^{be a}

pre-Hilbert

space with respect

to ~ ’,’)&#x3E;.

has in fact two

topologies;

^{as the}

quotient

^{of two closed}

subspaces

ofjf it has a

topology

^induced

by Jf,

^with respect ^to which it is

complete,

and furthermore it has a

topology

^induced

(since (

^x,

x ~ &#x3E;

^{0 on}

Jf’),

^with respect ^to which it need not to be

complete.

If the two

topologies

_{r~ and T~} ^are

equivalent

^{on then}

is

complete

^with respect ^{to the scalar}

product

^induced

by (’,’)&#x3E;;

^this

means that all the

physical

^{states are}

already

present ^{in X and that the} process of

taking

^the

quotient

^{does not}

require

^{a further}

completion.

The

physical

^{content of the}

theory

is therefore readable in Jf. Under

general conditions,

the boundedness of

11-

^{1 turns out to be a} necessary condition for the

completeness

^of

4. INFRARED

SINGULARITIES,

^{CLUSTER PROPERTY}

AND VACUUM STRUCTURE

The role and

physical implications

of infrared

singularities

ⁱⁿ

QFT’s

^has

attracted much attention

lately especially

ⁱⁿ connection with _gauge quantum field

theories,

^the confinement

mechanism,

^the

phenomenon

^of

^0-vacua,

^etc.

In the standard _case, in which the

Wightman

^functions

satisfy

^the

posi- tivity condition,

the situation is well understood ^at the

general

level: the translation invariance of the

Wightman

^functions

implies

^that the space- time translations are described

by unitary

operators

U(a)

and therefore the infrared

singularities

^{of the}

theory

^are rather mild. One can show in fact that for _any two local states

~P, 0

^the Fourier transform

U(x)1»

is a

(complex)

^{measure. This} property ^has strong consequences for the

phase

^{or vacuum structure of the}

theory

^{as shown}

by Araki, Hepp

^and

Ruelle

[27 ] :

^the

uniqueness (or

^non

uniqueness)

^{of the vacuum, which} guarantees ^the

irreducibility

^{of the field}

algebra,

^is

equivalent

^{to the vali-}

dity (or

^{the non}

validity)

^of the cluster property.

Thus, given

^{a set of}

Wight-

man functions which do not

satisfy

the cluster property, ^{the structure of} pure

phases (i.

^{e. theories with}

unique vacuum)

is obtained

by decomposing

the

given Wightman

functional into

positive

invariant functionals which

satisfy

^{the cluster property.}

The situation appears less clear in the indefinite metric case. It has sometimes been claimed that ^even in this case the non

validity

^of the cluster property ^{should be}

interpreted

^{as a}

sign

^{of the non}

uniqueness

^{of the}

Annales de l’Institut Henri Poincaré-Section A

vacuum, but the arguments ^{offered are not}

convincing.

^{A basic}

point

^is

that in the indefinite metric case the state content of the

theory

^{is not}

determined

by

^the

Wightman

^functions

alone,

^{since in}

general

^different

Hilbert _space structures are available and therefore the states which are

obtained

by taking

the closure of the local states

strongly depend

^{on the}

Hilbert

topology

^{one choses. In the}

general

^{case the}

question

of existence of more than one translation invariant state cannot be answered without reference to a

precise

^Hilbert space structure.

Actually,

^{as it will turn} out, the connection between the cluster property ^{and the}

uniqueness

^{of the}

vacuum does not hold in

general

^{and the}

problem

^{has to be}

investigated

anew.

To make the discussion more

precise

it is convenient to

classify

^the

infrared

singularities

^{in two classes.}

DEFINITION. 2014 We will _say that a set of

Wightman

functions have non

con, fining infrared singularities

^{if for} any ^two local states

~P, 0

^{the Fourier} transform

of ( ~P,

^{is a measure} in the

neighboorhood

^{of the}

light cone {q2

⁼

0}.

^{A set of}

Wightman

functions is said to exhibit

infrared singularities of

^the

confining

type ^if there are local states

BP,

^D such that the Fourier transform

of ( ~P,

^{is not a measure in the}

neighboorhood

of the

light cone {q2 =

⁰

}.

^I

One can prove that in theories with non

confining

^infrared

singularies

the cluster property may fail

by

^{a constant}

lim

[

⁺

03BBa)&#x3E; - B1(Xl) ] =

^{const ~ 0}

d ⁼ spacelike

and that this

implies

^the existence of more than one translation invariant

state

[27 ].

When

confining singularities

^are present ^{in the}

Wightman functions,

the

positivity

^{condition cannot} be satisfied and therefore the metric _opera- tor ~ ^{cannot be trivial}

[10].

^{In this case the} translation invariance of the

Wightman

^functions

only implies

^{that the} operators ^are

’1-unitary [22 ],

but

they

^{cannot be}

unitary [10 ]. Furthermore, they

^{are in}

general

^unbounded operators

[22 ],

^{with a common} dense domain which contains the local states. These

properties

^{make the}

analysis

^{of the} connection between the cluster property ^{and the}

uniqueness

^{of the vacuum much more delicate.}

For

example

^{if for a local state 0} the Fourier transform

of ( 1&#x3E;,

has support ^{at the}

origin,

^{one cannot conclude} that the state 0 is translation invariant

[28 ].

^This

unjustified

^{and in}

general

wrong conclusion underlies

most of the discussions of the

Schwinger model,

where the cluster property fails and nevertheless one can find a Hilbert _space structure of Sobolev type ^{in which the vacuum state} is

unique [29 ]. Actually,

^{this result can be}

viewed as a

special

^{case of a}

general

^theorem.

DEFINITION 3. 2014 ~ Hilbert _space structure with a

possibly dege-

nerate metric operator ~, ^{is said to be} Sobolev type

if ~03A80

^{= and there}

Vol. XXXIII, ^n° 3-1980.

264 G. MORCHIO AND F. STROCCHI

is a linear

correspondence

^{between a dense}

subspace

^{D c Jf and the}

space ~

^{such that}

1 )

^for

(0,

...,

in,

⁰

... )

^e~,

p(D = (~~.~,

^{is a Sobolev}

semi-norm

[30 on ~,

^{in momentum} space

2)

^for any

converging

ⁱⁿ

X,

the sequences of the n-th components, ^{are also}

converging [~7] in

^Jf.

If the

metric ~

^is

degenerate

^{one can}

always

^{define a new Hilbert} space structure

(1], ~), r~

⁼

(1

^-

% = _{(1 -}

where

Po

^is the pro-

jector

^{onto the}

subspace ~o = ~ x E ~; (y,

^{= 0}

Vy

^E

(see

^sect.

3).

PROPOSITION. Let

(t-~,

^{be a Hilbert} space ^structure of Sobolev type, with a

possibly degenerate

^metric operator 11, ^{and be the}

subspace

~ x (y,

⁼

0, Vy

If either of the

following

conditions holds

a) U(a)

^are

unitary

operators ^{on K}

b) ~~

then there is

only

^one translation invariant state in

.F

⁼

_~/~o of

^the

vacuum).

Proof

^{2014 We first show that}

a)

=&#x3E;

b).

^In

fact, if U(a)

^{are not}

only 1]-unitary

but also

unitary

operators ^{one has}

[U(a), 11] ] =

^{0 and therefore}

[U(a),

^{1 -}

Po] ]

^{== 0. i. e. c}

.ff Õ.

"-

Now, ^to prove that

b) implies

^the

uniqueness

^{of the vacuum in} Jf, ^let BP be ^a vector of

Jf,

^which

gives

^{rise to a} translation invariant

vector

in Jf.

Then,

^{one must have}

i. e.

Since, by

definition

~’o

is invariant under

translations,

^{the above}

equation

is

equivalent

^to

By

^condition

h), U(a)1&#x3E;

^E

.~o

^{and therefore} eq.

(3.2)

^can

only

be satisfied if

We will show that _eq.

(4 . 3)

^has

only

^the solution 03A6 = (03A60,

0, 0 ... ).

In

fact, by

^condition

2),

eq.

(4.3) implies (U(a)I&#x3E; -

⁼

0,

or, for the

corresponding fn

Since all

f

⁼

(0,0,

..., 0,

... ), belong

^to Sobolev spaces

they

^also

belong

^to

L2(R4n).

^Now it is well known that eq.

(3.4)

^{has no solution}

in

L 2(R 4n) (n &#x3E; 1),

different from zero.

Annales de l’Institut Henri Poincaré-Section A